Thomas Hangelbroek, Francis J. Narcowich, Christian Rieger and Joseph D. Ward An inverse theorem on bounded domains for meshless methods using localized bases 2014 http://arxiv.org/pdf/1406.1435v1
Aicke Hinrichs and Jens Oettershagen Optimal point sets for quasi-Monte Carlo integration of bivariate periodic functions with bounded mixed derivatives 2014 http://arxiv.org/pdf/1409.5894v1
Martin Huesmann Optimal transport between random measures Annales de l’Institut Henri Poincaré (B) 2014 http://arxiv.org/abs/1206.3672
Abstract: We study couplings q∙ of two equivariant random measures λ∙ and μ∙ on a Riemannian manifold (M,d,m). Given a cost function we ask for minimizers of the mean transportation cost per volume. In case the minimal/optimal cost is finite and λω≪m we prove that there is a unique equivariant coupling minimizing the mean transportation cost per volume. Moreover, the optimal coupling is induced by a transportation map, q∙=(id,T)∗λ∙. We show that the optimal transportation map can be approximated by solutions to classical optimal transportation problems on bounded regions. In case of Lp−cost the optimal transportation cost per volume defines a metric on the space of equivariant random measure with unit intensity.
2013
Benjamin Berkels, Tom Fletcher, Behrend Heeren, Martin Rumpf and Benedikt Wirth Discrete geodesic regression in shape space In Anders Heyden, Fredrik Kahl, Carl Olsson, Magnus Oskarsson, Xue-Cheng Tai, editor, Energy Minimization Methods in Computer Vision and Pattern Recognition, Volume 8081 of Lecture Notes in Computer Science
page 108-122.
Publisher: Springer International
2013 http://dx.doi.org/10.1007/978-3-642-40395-8_9
Abstract: A new approach for the effective computation of geodesic re- gression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity mea- sures between given two- or three-dimensional input shapes and corre- sponding shapes along the regression curve. The proposed method is based on a variational time discretization of geodesics. Curves in shape space are represented as deformations of suitable reference shapes, which renders the computation of a discrete geodesic as a PDE constrained optimization for a family of deformations. The PDE constraint is de- duced from the discretization of the covariant derivative of the velocity in the tangential direction along a geodesic. Finite elements are used for the spatial discretization, and a hierarchical minimization strategy together with a Lagrangian multiplier type gradient descent scheme is implemented. The method is applied to the analysis of root growth in botany and the morphological changes of brain structures due to aging.
Michael Griebel and Helmut Harbrecht A note on the construction of L-fold sparse tensor product spaces Constructive Approximation, 38(2): 235-251 2013 http://dx.doi.org/10.1007/s00365-012-9178-7
Abstract: In the present paper, we consider the construction of general sparse tensor product spaces in arbitrary space dimensions when the single subdomains are of different dimensionality and the associated ansatz spaces possess different approximation properties. Our theory extends the results from Griebel and Harbrecht (Math. Comput., 2013) for the construction of two-fold sparse tensor product space to arbitrary L-fold sparse tensor product spaces.
Martin Huesmann and Karl-Theodor Sturm Optimal transport from Lebesgue to Poisson The Annals of Probability, 41(4): 2426-2478 2013 http://dx.doi.org/10.1214/12-AOP814
Abstract: This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite. Moreover, we give precise conditions for the latter which demonstrate a sharp threshold at d=2d=2. The cost will be defined in terms of an arbitrary increasing function of the distance.
The coupling will be realized by means of a transport map (“allocation map”) which assigns to each Poisson point a set (“cell”) of Lebesgue measure 1. In the case of quadratic costs, all these cells will be convex polytopes.